An Efficient Algorithm of Convex Hull for Very Large Planar Point Set

Traditional geometric algorithms are often presented as if input imprecision does not exist, even though it is often unavoidable. We argue that in some cases, it may be desirable for geometric algorithms to treat this imprecision as an explicit component of the input, and to reflect this imprecision in the output. Starting with three problems from computational geometry whose inputs are planar point sets (Voronoi diagrams, convex hulls, and smallest bounding discs), we recast these as problems where each input point’s location is imprecise, but known to lie within a particular region of uncertainty. Where algorithms to solve each of the original problems produce a single geometric object as output, the algorithms that we present typically produce either guaranteed or possible output objects. A guaranteed object represents qualities that can be guaranteed for every point set that is consistent with the uncertain regions, and a possible object represents qualities that exist for at least one such point set. By dealing with input imprecision explicitly, these guaranteed and possible objects can represent a more accurate view of what can be reliably inferred from the input data.

Abstract. Erdös asked whether every large enough set of points in general position in the plane contains six points that form a convex 6-gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6-gon. To this end a fast incremental algorithm for finding such 6-gons was designed and implemented and a heuristic search approach was used to find promising sets. The experiments also led to two observations that might be useful in proving that large sets always contain an empty convex 6-gon.

In this paper, we developed an auto-focus image quality evaluation function from the histogram feature function (HFF) of a high-speed camera. Based on this function, we then proposed a fast auto-focus search algorithm. We further demonstrated that the auto-focus image quality evaluation function we proposed requires fewer computational resources, while providing a more effective focusing area and more moderate sensitivity, compared to the traditional evaluation function. In contrast to the traditional climbing method, our proposed fast auto-focusing search algorithm also can spot the focus point in a shorter time, effectively reducing the round trips of the focusing motor and improving the focusing accuracy.

This study examines various algorithms for computing the convex hull of a set of n points in a d-dimensional space. Convex hulls are fundamental in computational geometry and are applied in computer graphics, pattern recognition, and computational biology. Such convex hulls can also be useful in symmetry problems. For instance, when points are arranged symmetrically, the convex hull is also likely to be symmetrically shaped, which can be useful for object recognition in computer vision or pattern recognition. The focus is primarily on two-dimensional algorithms, including well-known methods like Gift Wrapping, Graham Scan, Divide and Conquer, QuickHull, TORCH, Kirkpatrick–Sediel, and Chan’s algorithms. These algorithms vary in terms of time complexity and scalability to higher dimensions. This study is extended to three-dimensional convex hull algorithms, such as NAW, randomized insertion, and parallelized versions, such as CudaHull and CudaChain. This study aimed to elucidate the operational principles, step-by-step procedures, and comparative time complexities of each algorithm. The implementation in Python facilitates a detailed comparison of the algorithmic performance through stepwise analysis and graphical outputs. The ultimate goal is to provide insights into the strengths and weaknesses of each algorithm under various scenarios, thereby offering a comprehensive guide for practical implementation.

This paper introduced points and directed line segment relation judgment method, the characteristics of generation and Graham method using the original convex hull generation algorithm of convex hull discrete points of the convex hull, an improved algorithm for planar discrete point set is proposed. The main idea is to use quadrilateral to divide planar discrete point set into five blocks, and then by judgment in addition to the four district quadrilateral internally within the point is in a convex edge. The result shows that the method is relatively simple program, high computational efficiency.

A chain of a set P of n points in the plane is a chain of the dominance order on P. A k-chain is a subset C of P that can be covered by k chains. A k-chain C is a maximum k-chain if no other k-chain contains more elements than C. This paper deals with the problem of finding a maximum k-chain of P in the cardinality and in the weighted case. Using the skeleton S(P) of a point set P introduced by Viennot we describe a fairly simple algorithm that computes maximum k-chains in time O(kn log n) and linear space. The basic idea is that the canonical chain partition of a maximum (k-1)-chain in the skeleton S(P) provides k regions in the plane such that a maximum k-chain for P can be obtained as the union of a maximal chain from each of these regions. By the symmetry between chains and antichains in the dominance order we may use the algorithm for maximum k-chains to compute maximum k-antichains for planar points in time O(kn log n). However, for large k one can do better. We describe an algorithm computing maximum k-antichains (and, by symmetry, k-chains) in time O((n2 k) log n) and linear space. Consequently, a maximum k-chain can be computed in time O(n3/2 log n) for arbitrary k. The background for the algorithms is a geometric approach to the Greene--Kleitman theory for permutations. We include a skeleton-based exposition of this theory and give some hints on connections with the theory of Young tableaux. The concept of the skeleton of a planar point set is extended to the case of a weighted point set. This extension allows to compute maximum weighted k-chains with an algorithm that is similar to the algorithm for the cardinality case. The time and space requirements of the algorithm for weighted k-chains are O(2kn log(2kn)) and O(2kn), respectively.

Triangulating with high connectivity

Advances in sensors and cameras allow current research in convex hull algorithms to focus on defining methods capable of processing a big set of points. Typically, in most of these algorithms, the orientation function needs around five sums and two multiplications. In this paper, we propose SymmetricHull, a novel algorithm that, unlike the related ones, only performs two comparisons per point, discarding points with a low probability of belonging to the convex hull. Our algorithm takes advantage of the symmetric geometry of convex hulls in 2D spaces and relies on the convexity principle to get convex hulls, without needing further calculations. Our experiments show that SymmetricHull achieves good results, in terms of time and number of necessary operations, resulting especially efficient with sets of points between \(10^4\) and \(10^7\). Given that our datasets are organized by quadrants, the features of our algorithm can be summarized as follows: (1) a fast point discard based on known points with a good chance to be part of the convex hull, (2) a lexicographic sort of points with a high probability of belonging to the convex hull, and (3) a simple slope analysis to verify whether a point is within the convex hull or not.

Our goal is the registration of multiple 3D point clouds obtained from LIDAR scans of underground mines. Such a capability is crucial to the surveying and planning operations in mining. Often, the point clouds only partially overlap and initial alignment is unavailable. Here, we propose an interactive user-assisted point cloud registration system. Guided by the system, the user's role is simply to identify and search for overlapping regions across the point clouds. Specifically, given two point sets, the user clicks on a point in one set, then simply hovers the mouse on the other set to find a matching point. Each mouse position gives rise to a translation, and our system instantly optimises the rotation that aligns the point clouds. Assuming that each individual point set is horizontally levelled with the ground by the level compensator on the LIDAR device, given a candidate 3D translation only one angular parameter needs to be estimated to rotationally align two 3D point sets. We propose a fast rotation search algorithm that delivers globally optimal results in real time. Our method conducts branch-and-bound optimisation with a novel bounding function whose evaluation amounts to simple sorted array operations. This provides smooth and accurate feedback to the user's search for overlapping regions. Our system is intuitive and helps to accelerate the registration of multiple scans.

Higher order digital nets are special classes of point sets for quasi-Monte Carlo rules which achieve the optimal convergence rate for numerical integration of smooth functions. An explicit construction of higher order digital nets was proposed by Dick, which is based on digitally interlacing in a certain way the components of classical digital nets whose number of components is a multiple ds of the dimension s. In this paper we give a fast computer search algorithm to find good classical digital nets suitable for interlaced components by using polynomial lattice point sets. We consider certain weighted Sobolev spaces of smoothness of arbitrarily high order, and derive an upper bound on the mean square worst-case error for digitally shifted higher order digital nets. Employing this upper bound as a quality criterion, we prove that the component-by-component construction can be used efficiently to find good polynomial lattice point sets suitable for interlaced components. Through this approach we are able to get some tractability results under certain conditions on the weights. Fast construction using the fast Fourier transform requires the construction cost of \(O(dsN \log N)\) operations using O(N) memory, where N is the number of points and s is the dimension. This implies a significant reduction in the construction cost as compared to higher order polynomial lattice point sets. Numerical experiments confirm that the performance of our constructed point sets often outperforms those of higher order digital nets with Sobol’ sequences and Niederreiter-Xing sequences used for interlaced components, indicating the usefulness of our algorithm.

We introduce a new measure for planar point sets S that captures a combinatorial distance that S is from being a convex set: The reflexivity rho(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study considers also some closely related quantities, such as the convex cover number kappa_c(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number kappa_p(S), which is given by the smallest number of convex chains with pairwise-disjoint convex hulls that cover S. We have proved that it is NP-complete to determine the convex cover or the convex partition number and have given logarithmic-approximation algorithms for determining each.

We show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon}\right)$ for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane and “fat” triangular ranges and for point sets in $\boldsymbol{R}^3$ and axis-parallel boxes; these are the first known nontrivial bounds for these range spaces. Our technique also yields improved bounds on the size of $\varepsilon$-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of $\varepsilon$-nets of size $O\left(\frac{1}{\varepsilon}\log\log\log\frac{1}{\varepsilon}\right)$ for the dual range space of “fat” regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich or of Even, Rawitz, and Shahar, we obtain improved approximation factors (computable in expected polynomial time by a randomized algorithm) for the hitting set or the set cover problems associated with the corresponding range spaces.

Block motion estimation using full search is computationally intensive. Previously proposed fast algorithms reduce the computation by limiting the number of searching locations. This is accomplished at the expense of less accuracy of motion estimation and gives rise to an appreciably higher mean squared error (MSE) for motion compensated images. We present a new fast and efficient search algorithm for block motion estimation that produces better quality performance and less computational time compared with a three-step search (TSS) algorithm. The proposed algorithms are based on the ideas of the dithering pattern for pixel decimation, multiple-candidate for pixel-decimation-based full search, and center-based distribution of the motion vector. From the experimental results, the proposed algorithm is superior to the TSS in both quality performance (about 0.2 dB) and computational complexity (about half).

This paper presents a fast efficient codeword search algorithm for vector quantization (VQ) based on the Adaptive Multi-Rate Wideband (AMR-WB) speech codec. AMR-WB is the speech codec standardized by the 3GPP for the GSM and WCDMA 3G system. It is also selected as the ITU-T G.722.2 standards. In AMR-WB speech codec VQ codeword search section, full search algorithm is used to find matching codeword .This requires a huge amount of computation. For large-size codebooks and high-dimensional vector of AMR-WB, computational complexity is significantly higher.The proposed fast codeword search method selects the codeword ,which has the least absolute error component in VQ codebook, as the initial matching codeword. It can help to reduce computational complexity, whereas the reconstructed speech quality almost has no dropping. The article introduces the basic principle of the fast codeword search algorithm in detail. Simulation results show that the algorithm based on AMR-WB speech codec, can reduce the time of vector quantization codebook search and the w-PESQ of nine modes is no significant dropping.

3-Points Convex Hull Matching (3PCHM) for fast and robust point set registration